CERTAIN deterministic models for the spatial spread of an epidemic or an advantageous gene among a population along a line can be analyzed in terms of the nonlinear convolution equation u (x)=(go u)* k (x) XER,(1.1) where go u is the composite of g and u, go u (x)= g (u (x)), and* denotes convolution,(* if/(x)= JR< f>(x-y) lf;(y) dy. In the epidemic model, g typically has the form g (x)= a (l-ex), a some constant (o:> 1), cf. Diekmann [1], while in the genetic model g is given by g (x)=[o: x2+/h (l-x)]/[o: x2+ 2/3x (l-x)+ y (l-x) 2], o:,{3, and y positive constants, cf. Weinberger [2]. In both cases, k is a nonnegative function, normalized such that Jak (x) dx= 1. We consider (1.1) under the following hypotheses: